Penalisation of the symmetric random walk by several functions of the supremum

TL;DR

This paper investigates how penalizing a symmetric random walk by various functions of its maximum alters its behavior, revealing significant differences under new probability measures despite similar penalization functions.

Contribution

It demonstrates that even closely related penalization functions can lead to markedly different behaviors of the process under the modified probabilities.

Findings

Different penalizations produce distinct process behaviors.

Close penalization functions can have divergent effects.

The process's properties change significantly under new measures.

Abstract

In this paper, we penalised the standard random walk by several functions of its maximum. The aim is to show that in spite of very close penalisation functions, under the new probabilities, the canonical process behaves very differently.